Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.80·2-s − 5.52·3-s + 15.0·4-s + 18.8·5-s + 26.5·6-s + 0.563·7-s − 33.8·8-s + 3.56·9-s − 90.3·10-s − 21.9·11-s − 83.2·12-s + 90.1·13-s − 2.70·14-s − 103.·15-s + 42.1·16-s − 68.5·17-s − 17.1·18-s + 29.2·19-s + 283.·20-s − 3.11·21-s + 105.·22-s − 171.·23-s + 187.·24-s + 228.·25-s − 432.·26-s + 129.·27-s + 8.48·28-s + ⋯
L(s)  = 1  − 1.69·2-s − 1.06·3-s + 1.88·4-s + 1.68·5-s + 1.80·6-s + 0.0304·7-s − 1.49·8-s + 0.132·9-s − 2.85·10-s − 0.601·11-s − 2.00·12-s + 1.92·13-s − 0.0516·14-s − 1.79·15-s + 0.658·16-s − 0.978·17-s − 0.224·18-s + 0.352·19-s + 3.16·20-s − 0.0323·21-s + 1.02·22-s − 1.55·23-s + 1.59·24-s + 1.83·25-s − 3.26·26-s + 0.923·27-s + 0.0572·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + 4.80T + 8T^{2} \)
3 \( 1 + 5.52T + 27T^{2} \)
5 \( 1 - 18.8T + 125T^{2} \)
7 \( 1 - 0.563T + 343T^{2} \)
11 \( 1 + 21.9T + 1.33e3T^{2} \)
13 \( 1 - 90.1T + 2.19e3T^{2} \)
17 \( 1 + 68.5T + 4.91e3T^{2} \)
19 \( 1 - 29.2T + 6.85e3T^{2} \)
23 \( 1 + 171.T + 1.21e4T^{2} \)
29 \( 1 - 9.36T + 2.43e4T^{2} \)
31 \( 1 - 244.T + 2.97e4T^{2} \)
37 \( 1 - 213.T + 5.06e4T^{2} \)
41 \( 1 + 191.T + 6.89e4T^{2} \)
47 \( 1 + 341.T + 1.03e5T^{2} \)
53 \( 1 + 488.T + 1.48e5T^{2} \)
59 \( 1 - 319.T + 2.05e5T^{2} \)
61 \( 1 + 51.1T + 2.26e5T^{2} \)
67 \( 1 + 733.T + 3.00e5T^{2} \)
71 \( 1 + 417.T + 3.57e5T^{2} \)
73 \( 1 - 35.6T + 3.89e5T^{2} \)
79 \( 1 - 32.5T + 4.93e5T^{2} \)
83 \( 1 + 73.6T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 309.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.558959161369235616485322322983, −8.055640391653353082869864794004, −6.66499154758209743975553419439, −6.24249856027295632024338323149, −5.79376935985151122461053136627, −4.65864754523927096471986533077, −2.87837255021818560163952190096, −1.83679200927478058004579943903, −1.13547548158704060747843006525, 0, 1.13547548158704060747843006525, 1.83679200927478058004579943903, 2.87837255021818560163952190096, 4.65864754523927096471986533077, 5.79376935985151122461053136627, 6.24249856027295632024338323149, 6.66499154758209743975553419439, 8.055640391653353082869864794004, 8.558959161369235616485322322983

Graph of the $Z$-function along the critical line